Commuting self-adjoint extensions of symmetric operators defined from the partial derivatives

We consider the problem of finding commuting self-adjoint extensions of the partial derivatives {(1/i)(partial derivative/partial derivativex(j)):j=1, ...,d} with domain C-c(infinity)(Omega) where the self-adjointness is defin ed relative to L-2(Omega), and Omega is a given open subset of R-d. The mea sure on Omega is Lebesgue measure on R-d restricted to Omega. The problem o riginates with Segal and Fuglede, and is difficult in general. In this pape r, we provide a representation-theoretic answer in the special case when Om ega =Ix Omega (2) and I is an open interval. We then apply the results to t he case when Omega is a d cube, I-d, and we describe possible subsets Lambd a subset ofR(d) such that {e(lambda)\(d)(I):lambda is an element of Lambda} is an orthonormal basis in L-2(I-d). (C) 2000 American Institute of Physic s. [S0022- 2488(00)02712-2].